Kernels are valuable tools in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers the links between them, as far as they are related to kernel techniques. It addresses non-expert readers and focuses on practical guidelines for using kernels in applications.

We review recent methods for learning with positive definite kernels. All these methods formulate learning and estimation problems as linear tasks in a reproducing kernel Hilbert space (RKHS) associated with a kernel. We cover a wide range of methods, ranging from simple classifiers to sophisticated methods for estimation with structured data.

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Abstract. Radial basis functions are well-known and successful tools for the interpolation of data in many dimensions. Several radial basis functions of compact support that give rise to nonsingular interpolation problems have been proposed, and in this paper we study a new, larger class of smooth radial functions of compact support which contains other compactly supported ones that were proposed earlier in the literature. 1.

In this paper we provide a framework for studying the approximation order resulting from using strictly positive definite kernels to do generalized Hermite interpolation and approximation on a compact Riemannian manifold. We apply this framework to obtain explicit estimates in cases of the circle and 2-sphere. In addition, we provide a technique for constructing strictly positive definite spherical functions out of radial basis functions, and we use it to make a spherical function that is locally supported.

. We show how conditionally negative definite functions on spheres coupled with strictly completely monotone functions (or functions whose derivative is strictly completely monotone) can be used for Hermite interpolation. The classes of functions thus obtained have the advantage over the strictly positive definite functions studied in [17] that closed form representations (as opposed to series expansions) are readily available. Furthermore, our functions include the historically significant spherical multiquadrics. Numerical results are also presented. AMS classification: 41A05, 41A63, 42A82. Key words and phrases: Spherical interpolation, Hermite interpolation, Radial basis functions. 1. Introduction In 1975 R. Hardy mentioned the possibility of using multiquadric basis functions for Hermite interpolation (see [10], or the survey paper [11]). This problem, however, was not further investigated until the paper [29] by Wu appeared. Since then, the interest in this topic seems to have ...

A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). A pseudometric on the space of probability measures can be defined as the distance between distribution embeddings: we denote this as γk, indexed by the kernel function k that defines the inner product in the RKHS. We present three theoretical properties of γk. First, we consider the question of determining the conditions on the kernel k for which γk is a metric: such k are denoted characteristic kernels. Unlike pseudometrics, a metric is zero only when two distributions coincide, thus ensuring the RKHS embedding maps all distributions uniquely (i.e., the embedding is injective). While previously published conditions may apply only in restricted circumstances (e.g., on compact domains), and are difficult to check, our conditions are straightforward and intuitive: integrally strictly positive definite kernels are characteristic. Alternatively, if a bounded continuous kernel is translation-invariant on R d, then it is characteristic if and only if the support of its Fourier transform is the entire R d.

Embeddings of probability measures into reproducing kernel Hilbert spaces have been proposed as a straightforward and practical means of representing and comparing probabilities. In particular, the distance between embeddings (the maximum mean discrepancy, or MMD) has several key advantages over many classical metrics on distributions, namely easy computability, fast convergence and low bias of finite sample estimates. An important requirement of the embedding RKHS is that it be characteristic: in this case, the MMD between two distributions is zero if and only if the distributions coincide. Three new results on the MMD are introduced in the present study. First, it is established that MMD corresponds to the optimal risk of a kernel classifier, thus forming a natural link between the distance between distributions and their ease of classification. An important consequence is that a kernel must be characteristic to guarantee classifiability between distributions in the RKHS. Second, the class of characteristic kernels is broadened to incorporate all strictly positive definite kernels: these include non-translation invariant kernels and kernels on non-compact domains. Third, a generalization of the MMD is proposed for families of kernels, as the supremum over MMDs on a class of kernels (for instance the Gaussian kernels with different bandwidths). This extension is necessary to obtain a single distance measure if a large selection or class of characteristic kernels is potentially appropriate. This generalization is reasonable, given that it corresponds to the problem of learning the kernel by minimizing the risk of the corresponding kernel classifier. The generalized MMD is shown to have consistent finite sample estimates, and its performance is demonstrated on a homogeneity testing example. 1

Dedicated to Yum-Tong Siu on the occasion of his sixtieth birthday 0. Introduction. We introduce a family of positivity conditions for Hermitian symmetric functions, establish basic properties, and connect the ideas with complex geometry. Let M be a complex manifold, and let M ′ denote its complex conjugate manifold. In this paper M will typically be either C n or the total space of a holomorphic line bundle over a compact complex manifold.

Black--box models based on kernels K are written as mappings of the form j K(x j ; x) that are intended to reproduce observational input/output data pairs (x j ; y j ) in the sense F (x j ) y j . Such functions have been studied in a general mathematical context for quite some time, and this contribution reviews part of the known facts and provides links to a subset of the background literature. Special emphasis is given to questions of optimality and complexity within the context of black--box modelling.